\(\Omega\) abstract set representing the sample space of a random experiment. The elements in \(\omega \in \Omega\) are the possible outcomes of the experiment.
\(\mathcal{P}(\Omega)\): power set of \(\Omega\), the set of all possible subsets of \(\Omega\).
Most of subsets \(A, B, \dots\) will be thought as events.
Collection of subsets \(\mathcal{A}, \mathcal{B}, \dots\).
The empty set \(\emptyset\).
\(\mathcal{B}\) is a \(\sigma\)-field, usually connected with the sample space \(\Omega\).
\(\mathcal{B}(\mathbb{R})\) is the Borel \(\sigma\)-field of \(\mathbb{R}\).
\(\mathbb{P}\) is a probability measure function \(\mathbb{P}: \mathcal{B} \to [0,1]\).
\((\Omega, \mathcal{B}, \mathbb{P})\) is a probability space.
\({\color{red}{\sqcup}}\) is a shortcut to denote a disjoint union, for example writing \(A {\color{red}{\sqcup}} B\) means that the sets \(A\) and \(B\) are disjoint, while writing \(A {\color{red}{\cup}} B\) means that the sets \(A\) and \(B\) are not disjoint.
\(X\) denotes an univariate random variable.
Bold letters, \(\mathbf{X}_n = (X_1, X_2, \dots, X_i, \dots, X_n)\) denotes a random vector with \(n\) elements, where each \(X_i\) is a random variable. Instead, \(\mathbf{x}_n = (x_1, x_2, \dots, x_i, \dots, x_n)\) denotes a vector, where each \(x_i\) is a scalar.
\(\mathbf{X}\) denotes a random matrix. More precisely, \(\mathbf{X} = (\mathbf{X}_1, \mathbf{X}_2, \dots, \mathbf{X}_j, \dots, \mathbf{X}_k)\), where \(\mathbf{X}_j\) is a random vector related to the column \(j\) with \(n\)-elements. In matrix context, the index \(i\) denotes a generic row, while \(j\) a generic column.